Low Power on Shortwave?

Yes, in the USA, shortwave low power stations may operate in the 22 meter shortwave band from 13.553 MHz to 13.567 MHz, with a field strength limit of 10,000 uV (microvolts) per meter at 30 meters from the antenna.
About 1.8 mW to 3 mW would be needed, depending on the antenna. Unlike low power AM medium wave, however, there is no limitation on the size or type of antenna, feedline or ground for shortwave low power stations.
It would be an almost unknown operation here in the US, as very few people own shortwave radio tuners and even fewer ever listen. In other parts of the world that might be different.
Here are some schematics, plans, illustrations and ordering information for low power shortwave transmitters:
---Amplified modulation section design---No-coil design---Hartley Oscillator Kit info---Talking Pixie modified ham transmitter, and filterTry Amazon Prime 30-Day Free Trial